Problem 11 - Algebra 1: Concepts and Skills

Question

Check whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{c} {-15x+7y=1} \\ { 3x-y=1} \end{array} \quad(3,5) $$

Step-by-Step Solution

Answer

No, the ordered pair (3,5) is not a solution for the system of provided equations.

Step 1: Identify the Given Values

Identify the given values from the ordered pair. The ordered pair (3,5) provides the values of x and y. Here, \(x = 3\) and \(y = 5\).

Step 2: Substitute x and y in the First Equation

Substitute \(x = 3\) and \(y = 5\) in the first equation. This gives: \(-15x+7y = -15(3) + 7(5) = -45 + 35 = -10\). The left-hand side of the first equation is \(-10\), which is not equal to 1, therefore the ordered pair (3,5) is not a solution for first equation.

Step 3: Substitute x and y in the Second Equation

Although the first equation does not hold true with the given ordered pair, let's substitute \(x = 3\) and \(y = 5\) in the second equation as well for completeness: \(3x - y = 3(3) - 5 = 9 - 5 = 4\). The left-hand side of the equation is \(4\), which is not equal to 1, therefore the ordered pair (3,5) is not a solution for second equation either.

Key Concepts

solution of equations

When facing a system of linear equations, such as the two provided in the exercise, you want to determine if there are values for the variables that make all equations true at the same time. These values are referred to as the \(x\) and \(y\) values in this particular system.

To find out if a specific pair is a solution, you replace \( x \) and \( y \) with the given numbers in both equations. The goal is to see if the left-hand side of each equation equals the right-hand side after substituting. If so, then the ordered pair is a solution for that equation.

• For example, using the pair (3,5) and substituting into both equations showed that neither resulted in a true statement, as the left-hand side was different from the right-hand side.
• This means (3,5) is not a solution for this system because it doesn't satisfy either equation.

ordered pairs

An ordered pair in mathematics typically refers to a set of two numbers used to represent points in a coordinate system. In systems of equations, ordered pairs correspond to the \(x\) and \(y\) coordinates that might satisfy both equations simultaneously.

The ordered pair is written in the form \((x, y)\). When testing if an ordered pair is a solution, you check if this \(x, y\) combo makes both equations true. Since the goal in the given exercise was to ascertain the validity of the pair (3,5), you substitute \(x = 3\) and \(y = 5\) into both equations and solve them. If both equations are true, the ordered pair is a solution of the system.

• With the ordered pair (3,5), neither equation checked out, indicating that (3,5) is not a solution for the system provided.
• Effective analysis involves careful examination of each substitution and resulting simplification.

substitution method

The substitution method is a technique often used to solve systems of two linear equations. The method involves solving one of the equations for one variable, then substituting that expression into the other equation.

This method was not fully employed in the existing solution, which simply tested if a given ordered pair satisfies both equations. To implement the substitution method, you would need to:

• Solve one of the equations to express one variable in terms of the other, for example, lift \( y \) from the second equation, \( 3x - y = 1 \) to find \( y = 3x - 1 \).
• Substitute \( y = 3x - 1 \) into the first equation and solve for \( x \).
• Use the value of \( x \) to find \( y \) using your expression from step one.
Userful in various scenarios, the substitution method helps you find solutions of the equations beyond trial and error with ordered pairs, providing exact solutions when possible.